Abstract
In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0+a 1 x 1+⋯+a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form (a 0+a 1 x 1+⋯+a n x n )/(b 0+b 1 x 1+⋯+b n x n ) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo’s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo’s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.
Research partially supported by CNPq Prosul Proc. no. 490333/04-4 (Brazil).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Billionnet, A.: Approximation algorithms for fractional knapsack problems. Operations Research Letters 30(5), 336–342 (2002)
Bitran, G.R., Magnanti, T.L.: Duality and sensitivity analysis for fractional programs. Operations Research 24, 675–699 (1976)
Carlson, J., Eppstein, D.: The weighted maximum-mean subtree and other bicriterion subtree problems. In: ACM Computing Research Repository(2005) cs.CG/0503023
Chen, D.Z., Daescu, O., Dai, Y., Katoh, N., Wu, X., Xu, J.: Efficient algorithms and implementations for optimizing the sum of linear fractional functions, with applications. Journal of Combinatorial Optimization 9, 69–90 (2005)
Dantzig, G.B., Blattner, W., Rao, M.R.: Finding a cycle in a graph with minimum cost to time ratio with applications to a ship routing problem. In: Rosensthiel, P. (ed.) Theory of Graphs: Int. Symposium, Dunod, Paris, pp. 77–84 (1967)
Dasdan, A., Gupta, R.K.: Faster maximum and minimum mean cycle algorithms for system-performance analysis. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 17(10), 889–899 (1998)
Dasdan, A., Irani, S.S., Gupta, R.K.: Efficient algorithms for optimum cycle mean and optimum cost to time ratio problems. In: Proceedings of the 36th ACM/IEEE Conference on Design Automation, pp. 37–42 (1999)
Dinkelbach, W.: On nonlinear fractional programming. Management Science 13, 492–498 (1967)
Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-Completeness. W.H. Freeman and Co., New York (1979)
Gubbala, P., Raghavachari, B.: Finding k-connected subgraphs with minimum average weight. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 212–221. Springer, Heidelberg (2004)
Hashizume, S., Fukushima, M., Katoh, N., Ibaraki, T.: Approximation algorithms for combinatorial fractional programming problems. Mathematical Programming 37, 255–267 (1987)
Jagannathan, R.: On some properties of programming problems in parametric form pertaining to fractional programming. Management Science 12, 609–615 (1966)
Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)
Jain, K., Măndoiu, I., Vazirani, V.V., Williamson, D.P.: A primal-dual schema based approximation algorithm for the element connectivity problem. Journal of Algorithms 45(1), 1–15 (2002)
Katoh, N.: A fully polynomial-time approximation scheme for minimum cost-reliability ratio problems. Discrete Applied Mathematics 35(2), 143–155 (1992)
Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. Journal of the Association for Computing Machinery 41(2), 214–235 (1994)
Klau, G., Ljubi, I., Mutzel, P., Pferschy, U., Weiskircher, R.: The fractional prize collecting Steiner tree problem on trees. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 691–702. Springer, Heidelberg (2003)
Lawler, E.L.: Optimal cycles in doubly weighted directed linear graphs. In: Rosensthiel, P. (ed.) Theory of Graphs: Int. Symposium, Dunod, Paris, pp. 209–214 (1967)
Megiddo, N.: Combinatorial optimization with rational objective functions. Mathematics of Operations Research 4(4), 414–424 (1979)
Orlin, J.B., Ahuja, R.K.: New scaling algorithms for the assignment and minimum mean cycle problems. Mathematical Programming 54(1), 41–56 (1992)
Radzik, T.: Newton’s method for fractional combinatorial optimization. In: Proc. 33rd Annual Symposium on Foundations of Computer Science, pp. 659–669 (1992)
Shigeno, M., Saruwatari, Y., Matsui, T.: An algorithm for fractional assignment problems. Discrete Applied Mathematics 56(2–3), 333–343 (1995)
Skiscim, C.C., Palocsay, S.W.: The complexity of minimum ratio spanning tree problems. Journal of Global Optimization 30, 335–346 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Correa, J.R., Fernandes, C.G., Wakabayashi, Y. (2006). Approximating Rational Objectives Is as Easy as Approximating Linear Ones. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_33
Download citation
DOI: https://doi.org/10.1007/11785293_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35753-7
Online ISBN: 978-3-540-35755-1
eBook Packages: Computer ScienceComputer Science (R0)